In mathematics, a measurable acting group is a special group that acts on some space in a way that is compatible with structures of measure theory. Measurable acting groups are found in the intersection of measure theory and group theory, two sub-disciplines of mathematics. Measurable acting groups are the basis for the study of invariant measures in abstract settings, most famously the Haar measure, and the study of stationary random measures.

This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these template messages) The topic of this article may not meet Wikipedia's general notability guideline. (September 2018) This article relies largely or entirely on a single source. (September 2018)

Let ${\displaystyle (G,{\mathcal {G}},\circ )}$ be a measurable group, where ${\displaystyle {\mathcal {G}}}$ denotes the ${\displaystyle \sigma }$-algebra on ${\displaystyle G}$ and ${\displaystyle \circ }$ the group law. Let further ${\displaystyle (S,{\mathcal {S}})}$ be a measurable space and let ${\displaystyle {\mathcal {A}}\otimes {\mathcal {B}}}$ be the product ${\displaystyle \sigma }$-algebra of the ${\displaystyle \sigma }$-algebras ${\displaystyle {\mathcal {A}}}$ and ${\displaystyle {\mathcal {B}}}$.

Let ${\displaystyle G}$ act on ${\displaystyle S}$ with group action

${\displaystyle \Phi \colon G\times S\to S}$

If ${\displaystyle \Phi }$ is a measurable function from ${\displaystyle {\mathcal {G}}\otimes {\mathcal {S}}}$ to ${\displaystyle {\mathcal {S}}}$, then it is called a measurable group action. In this case, the group ${\displaystyle G}$ is said to act measurably on ${\displaystyle S}$.

One special case of measurable acting groups are measurable groups themselves. If ${\displaystyle S=G}$, and the group action is the group law, then a measurable group is a group ${\displaystyle G}$, acting measurably on ${\displaystyle G}$.

• Kallenberg, Olav (2017). Random Measures, Theory and Applications. Probability Theory and Stochastic Modelling. Vol. 77. Switzerland: Springer. doi:10.1007/978-3-319-41598-7. ISBN 978-3-319-41596-3.

This group theory-related article is a stub. You can help Wikipedia by expanding it.

• v
• t
• e